Harmonic balance approach to limit cycles for nonlinear jerk equations

Abstract

The method of harmonic balance (HB) is employed to estimate the attributes of limit cycles of some nonlinear third-order (jerk) differential equations which are parity-invariant but not time-reversal-invariant. Two examples with cubic nonlinearities show that the HB method can give good values for the frequency and both the velocity and displacement amplitudes of a period one limit cycle.

Comparisons between harmonic balance and nonlinear output frequency response function in nonlinear system analysis

“…Such nonlinear systems range from models such as the simple the Duffing oscillator [14] to more complex models such as cracked rotors [15]. More applications of the harmonic balance method can be found in the study of the nonlinear response of airfoils [16]- [17], non-linear conservative systems [18], hysteretic two-degree-of-freedom systems [19], the third order (jerk) differential equations [20] and the Jeffcott rotor [21]. By using the HBM, some interesting phenomena unique to nonlinear systems have been observed, among which the most well-known is jump phenomenon where the response amplitude of a nonlinear oscillator changes suddenly at some critical value of the frequency of the excitation [13].…”

New Concepts in Electromagnetic Jerky Dynamics and Their Applications in Transient Processes of Electric Circuit

“…SLG jerky dynamics has transcend traditional physics. It generalized the classical mechanical concept (jerk) into the mathematical jerk function or jerk equation and discussed the regular [17,18], chaotic [3,4] solutions and the relative rules [11,19]. Some summarizations and remarks have been given in Ref.…”

He’s Variational Iteration Method for Nonlinear Jerk Equations: Simple but Effective

“…(1) by harmonic balance (HB) method [6,8], homotopy perturbation method (HPM) [9], perturbation method [10], Mickens iteration procedure [11] and homotopy analysis method [12]. The purpose of this paper is to apply variational iteration method which is useful for finding an periodic solution of non-linear jerk equations.…”